A mathematical study to select fractionation regimen based on physical dose distribution and the linear-quadratic model

Purpose: Hypofractionated irradiation is often used in precise radiotherapy instead of conventional multifractionated irradiation. We propose a novel mathematical method for selecting a hypofractionated or multifractionated irradiation regimen based on physical dose distribution adding to biologic consideration.

Methods and materials: The linear-quadratic model was used for the radiation effects on tumor and normal tissues, especially organs at risk (OARs). On the basis of the assumption that the OAR receives a fraction of the dose intended for the tumor, the minimization problem for the damage effect on the OAR was treated under the constraint that the radiation effect on the tumor is fixed.

Results: For an N-time fractionated irradiation regimen, the constraint of tumor lethality was described by an N-dimensional hypersphere. The total dose of the fractionated irradiations was considered for minimizing the damage effect on the OAR under the hypersphere condition. It was found that the advantage of hypofractionated or multifractionated irradiation therapies depends on the magnitude of the ratio of α/β parameters for the OAR and tumor in the linear-quadratic model and the ratio of the dose for the OAR and tumor.

Conclusions: Our mathematical method shows that multifractionated irradiation with a constant dose is better if the ratio of α/β for the OAR and tumor is less than the ratio of the dose for the OAR and tumor, whereas hypofractionated irradiation is better otherwise.